Xiuchuan Zhang

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This is Xiuchuan's personal website.
I plan to post some of my current learning and review notes on it.
If you have any questions or suggestions, welcome to comment in my posts.

Some review notes are from:
Complex Analysis

Complex Analysis

Introduction

Birth

• Cubic equations $x^3 = px + q$ always must be a solution
• Del Ferro (1465-1526) and Tartaglia (1499-1577), followed by Cardano (1501-1576), showed that the solution given by
$\displaystyle x = \sqrt[3]{\sqrt{\frac{q^2}{4}-\frac{p^3}{27}}+\frac{q}{2}}-\sqrt[3]{\sqrt{\frac{q^2}{4}-\frac{p^3}{27}}-\frac{q}{2}}$
• About 30 years after the discovery of this formula, Bombelli (1526-1572) consider $\displaystyle\frac{q^2}{4}-\frac{p^3}{27} < 0$
• It showed that perfectly real problems require complex arithmetic for their solution.

Properties and Definitions

• Complex numbers: expressions of the form $z = x + iy$, where
• x is called the real part of $z; x = \mathrm{Re} z$
• y is called the imaginary part of $z; y = \mathrm{Im} z$
• Set of complex numbers: $\mathbb{C}$ (the complex plane)
• Real numbers: subset of the complex numbers (those whose imaginary part is zero)
• The complex plane can be identified with $\mathbb{R}^{2}$

• $\mathrm{Re}(z + w) = \mathrm{Re}\ z + \mathrm{Re}\ w \ \mathrm{and} \ \mathrm{Im}(z + w) = \mathrm{Im}\ z + \mathrm{Im}\ w$
• Def: The modulus of the complex number z = x + iy is the length of the vector z:
• $\vert z\vert = \sqrt{x^2 + y^2}$
• $(x + iy) \cdot (u + iv) = (xu - yv) + i(xv + yu) \in \mathbb{C}$
• $\displaystyle \frac{x + iy}{u + iv} = \frac{xu + yv}{u^2 + v^2} + i\frac{yu - xv}{u^2 + v^2} \ (\mathrm{for} \ u + iv \neq 0)$

• If $z = x + iy$ then $\overline{z} = x − iy$ is the complex conjugate of z
• $\overline{\overline{z}} = z$
• $\overline{z + w} = \overline{z} + \overline{w}$
• $\vert z\vert = \vert \overline{z}\vert$
• $z\overline{z} = (x + iy)(x - iy) = x^2 + y^2 = \vert z\vert ^2$
• $\displaystyle \frac{1}{z} = \frac{\overline{z}}{z\overline{z}} = \frac{\overline{z}}{\vert z\vert ^2}$
• $\vert z \cdot w \vert = \vert z\vert \cdot \vert w\vert$
• $\displaystyle \overline{(\frac{z}{w})} = \frac{\overline{z}}{\overline{w}}$
• $\vert z\vert = 0 \ \mathrm{iff} \ z = 0$
• $−\vert z\vert ≤ \mathrm{Re}\ z ≤ \vert z\vert$
• $−\vert z\vert ≤ \mathrm{Im}\ z ≤ \vert z\vert$
• $\vert z + w\vert ≤ \vert z\vert + \vert w\vert \ \mathrm{(triangle \ inequality)}$
• $\vert z − w\vert ≥ \vert z\vert − \vert w\vert \ \mathrm{(reverse\ triangle\ inequality)}$
• The Fundamental Theorem of Algebra
If $a_{0}, a_{1}, \dots , a_{n}$ are complex numbers with $a_{n} \neq 0$ , then the polynomial
$\displaystyle \displaystyle p(z) = a_{n}z^{n} + a_{n-1}z^{n-1} + \dots + a_{1}z + a_{0}$
has n roots $z_{1}, z_{2}, \dots , z_{n} \ \mathrm{in} \ \mathbb{C}$
It can be factored as
$p(z) = a_{n}(z − z_{1})(z − z_{2})\dots(z − z_{n})$

Polar Representation

• $r = \vert z\vert$: the distance from the origin
• $θ$: the angle between the positive x-axis and the line segment from 0 to z
• $(r, θ)$ are the polar coordinates of z
• Relation between Cartesian and polar coordinates:
• $x = r cos θ$
• $y = r sin θ$
• $z = r(cos θ + i sin θ)$
• Def: The principal argument of z, called Arg z, is the value of θ for which $−π < θ ≤ π$.
• $arg z = \lbrace Arg z + 2πk : k = 0, ±1, ±2, . . .\rbrace,\ z \neq 0$
• Convenient notation: $e^{i\theta} = cos θ + i sin θ$
• the polar form of z: $z = r e^{i\theta}$
• $\vert e^{i\theta}\vert = 1$
• $\overline{e^{i\theta}\ =\ e^{-i\theta}}$
• $\displaystyle \frac{1}{e^{i\theta}}\ =\ e^{-i\theta}$
• $e^{i(\theta\ + \ \varphi)}\ = e^{i\theta}\ \cdot\ e^{i\varphi}$
• $arg(\overline{z})\ = \ -arg\ z$
• $arg(\frac{1}{z})\ = \ - arg\ z$
• $arg(z_1 z_2)\ = \ arg(z_1) \ + \ arg(z_2)$
• $(e^{i\theta})^n\ = \ e^{i\cdot n\theta}$
• $(cos θ + i sin θ)^n \ = \ cos(nθ)\ + \ i sin(nθ)$

Roots of Complex Numbers

• Def: Let $w$ be a complex number. An nth root of $w$ is a complex number $z$ such that $z^n = w$
• Use the polar form for $w$ and $\ z:\ w\ = \ \rho e^{i\varphi}\ \mathrm{and}\ z\ = \ r e^{i\theta}$
• $z^{n}\ = \ w \ : \ r^{n}e^{in\theta}\ = \ \rho e^{i\varphi}, \mathrm{so} r^n\ = \ \rho \ \mathrm{and} \ e^{in\theta} \ = \ e^{i\varphi}$
• $r = \sqrt[n]{\rho} \ \mathrm{and} \ n\theta \ = \ \varphi \ + \ 2k\pi, \ k \in \mathbb{Z}$
• $\displaystyle w^{\frac{1}{n}}\ = \ \sqrt[n]{\rho}\ e^{i(\frac{\varphi}{n} \ + \ \frac{2k\pi}{n})},\ k\ = \ 0, 1, \dots , \ n - 1$
• Def: The n_th roots of 1 are called the _nth roots of unity
• Since $1\ =\ 1\ e^{i\dot 0}$, we find that
%

Topology in the Plane

• Sets in the Complex Plane
• Circles and disks: center $z_0\ =\ x_0\ +\ i y_0$, radius $r$.
• $B_r(z_0) = {z \in \mathbb{C} : z$ has distance less than $r$ from $z_0}$ disk of radius $r$ , centered at $z_0$.
• $K_r(z_0) = {z \in \mathbb{C} : z$ has distance $r$ from $z_0}$ circle of radius $r$ , centered at $z_0$
• Measure distance
• so $B_r(z_0)\ =\ {z\ ∈\ \mathbb{C}\ :\ \vert z\ −\ z_0\vert \ <\ r}\ \mathrm{and}\ K_r(z_0)\ =\ {z ∈ \mathbb{C}\ :\ \vert z\ −\ z_0\vert\ =\ r}$
• Interior Points and Boundary Points
• Def: Let $E ⊂ \mathbb{C}$. A point $z_0$ is an interior point of $E$ if there is some $r > 0$ such that $B_r(z_0) ⊂ E$.
• Def: Let $E ⊂ \mathbb{C}$. A point $b$ is a boundary point of $E$ if every disk around $b$ contains a point in $E$ and a point not in $E$.
The boundary of the set $E ⊂ \mathbb{C}, ∂E,$ is the set of all boundary points of $E$.
• Open and Closed Sets
• A set $U ⊂ \mathbb{C}$ is open if every one of its points is an interior point.
• A set $A ⊂ \mathbb{C}$ is closed if it contains all of its boundary points
• Examples:
• ${z\ ∈\ \mathbb{C}\ :\ \vert z\ −\ z_0 \vert \ <\ r}\ \mathrm{and} {z\ ∈\ \mathbb{C}\ :\ \vert z\ −\ z_0 \vert\ >\ r}$ are open
• $\mathbb{C}$ and $∅$ are open
• ${z\ ∈\ \mathbb{C}\ :\ \vert z\ −\ z_0 \vert \ ≤\ r}\ \mathrm{and} {z\ ∈\ \mathbb{C}\ :\ \vert z\ −\ z_0 \vert\ =\ r}$ are closed
• $C$ and $∅$ are closed
• ${z\ ∈\ \mathbb{C}\ :\ \vert z\ −\ z_0 \vert \ <\ r}\ ∪ {z\ ∈\ \mathbb{C}\ :\ \vert z\ −\ z_0 \vert\ =\ r\ \mathrm{and Im} (z\ −\ z_0)\ >\ 0}$ is neither open nor closed
• Closure and Interior of a Set
• Def:: Let $E$ be a set in $\mathbb{C}$.
The closure of $E$ is the set $E$ together with all of its boundary points: $\overline{E}\ =\ E\ ∪\ ∂E$.
The interior of $E$ , $\circ{E}$ is the set of all interior points of $E$.
• Examples:
• $\overline{B_r(z_0)}\ = \ B_r(z_0)\ \cup \ K_r(z_0)\ = \ {z\ \in\ \mathbb{C}\ :\ \vert z\ -\ z_0 \vert \ \leq \ r}$
• $\overline{K_r(z_0)}\ = \ K_r(z_0)$
• $\overline{B_r(z_0)\ \setminus \ {z_0}}\ = \ {z\ \in\ \mathbb{C}\ :\ \vert z\ -\ z_0 \vert \ \leq \ r}$
• With $E\ =\ {z\ \in\ \mathbb{C}\ :\ \vert z\ -\ z_0 \vert \ \leq \ r}, \circ{E}\ = \ B_r(z_0)$
• With $E\ =\ K_r(z_0),\ \circ{E}\ = \ \emptyset$
• Connectedness
• Def: Two sets $X, Y$ in $\mathbb{C}$ are separated if there are disjoint open set $U, V$ so that $X\ ⊂\ U$ and $Y\ ⊂\ V$ . A set $W$ in $\mathbb{C}$ is connected if it is impossible to find two separated non-empty sets whose union equals $W$.
• Examples:
• $X\ =\ [0, 1)$ and $Y\ =\ (1, 2]$ are separated
• Choose $U\ =\ B_1(0)$ and $V\ =\ B_1(2)$ . Thus $X\ ∪\ Y\ =\ [0, 2]\ \setminus \ {1}$ is not connected
• Theorem: Let $G$ be an open set in $\mathbb{C}$ . Then $G$ is connected if and only if any two points in $G$ can be joined in $G$ by successive line segments.
• Def: A set $A$ in $\mathbb{C}$ is bounded if there exists a number $R\ >\ 0$ such that $A\ ⊂\ B_R(0)$ . If no such $R$ exists then $A$ is called unbounded.
• The Point at Infinity
• In $\mathbb{R}$ , there are two directions that give rise to $±∞$
• In $\mathbb{C}$ , there is only one $∞$ which can be attained in many directions.

Function

Complex Functions

• A function $f:\ A\ \to\ B$ is a rule that assigns to each element $A$ of exactly one element of $B$
• $f^n(z)$ (read: “Eff n”) is called the nth iterate of f

• Sequences and Limits of Complex Numbers
• Def: A sequence {sn} of complex numbers converges to $s ∈ \mathbb{C}$ if for every $ε > 0$ there exists an index $N ≥ 1$ such that
$% $
We write
$\displaystyle\lim_{x \to \infty}\ s_n\ =\ s.$
• Rules for Limits
1. Convergent sequences are bounded.
2. If ${s_n}$ converges to $s$ and ${t_n}$ converges to $t$ , then
• $s_n\ +\ t_n\ \to \ s\ +\ t$
• $s_n\ \dot \ t_n\ \to \ s\ \dot \ t$ (in particular: $a\ \dot \ s_n \ \to \ a\ \dot \ s$ for any $a \in \mathbb{C}$ )
• $\displaystyle \frac{s_n}{t_n}\ \to \ \frac{s}{t}$ , provided $t\ \neq \ 0$
• Facts
• A sequence of complex numbers, ${s_n}$, converges to $0$ iff the sequence ${\vert s_n \vert}$ of absolute values converges to $0$
• A sequence of complex numbers, ${s_n}$, with $s_n\ =\ x_n\ +\ i y_n$ , converges to $s\ =\ x\ +\ i y$ iff $x_n\ \to \ x$ and $y_n\ \to \ y$ as $n\ \to \ \infty$
• Squeeze Theorem: Suppose that ${r_n}$ , ${s_n}$ and ${t_n}$ are sequences of real numbers such that $r_n\ ≤\ s_n\ ≤\ t_n$ for all $n$ . If both sequences ${r_n}$ and ${t_n}$ converge to the same limit, $L$ , then the sequence ${s_n}$ converges to the limit $L$ as well.

• Theorem: A bounded, monotone sequence of real numbers converges

• Limits of Complex Functions
• Def: The complex-valued function $f(z)$ has limit $L$ as $z\ \to \ z_0$ if the values of $f(z)$ are near $L$ as $z\ \to \ z_0$ .
• Also: $\displaystyle \lim_{z \to z_0}\ f(z)\ = \ L$ if for all $ε\ >\ 0$ there exists $δ\ >\ 0$ such that $\vert f(z)\ −\ L\vert \ <\ ε$ whenever $0\ <\ \vert z\ −\ z_0\vert \ <\ δ$
• Facts
• If $f$ has a limit at $z_0$ then $f$ is bounded near $z_0$ .
• If $f(z)\ \to \ L$ and $g(z)\ \to \ M$ as $z\ \to \ z_0$ then
• $f(z)\ + \ g(z) \ \to \ L \ + \ M$ as $z\ \to \ z_0$
• $f(z)\ \dot \ g(z) \ \to \ L \ \dot \ M$ as $z\ \to \ z_0$
• $\displaystyle \frac{f(z)}{g(z)} \ \to \ \frac{L}{M}$ as $z\ \to \ z_0$, provided that $M\ \neq \ 0$
• Continuity
• Def: The function $f$ is continuous at $z_0$ if $f(z)\ \to \ f(z_0)$ as $z\ \to \ z_0$ .
• This definition implicitly says that:
• $f$ is defined at $z_0$
• $f$ has a limit as $z\ \to \ z_0$
• The limit equals $f(z_0)$

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